Let's face it I suck at IT work! Even simple tasks seem to take
a long time and require countless one hour calls to "hello my
name is Rikki" in Mumbai.

First some good news. I got the Linksys WRT54GL Wireless
Broadband Router. Ordered it on-line as its not the latest
Linksys/Cisco model found in the local Frys.

The goal is very simple. Connected two routers together in
cascade, separated by 100 feet of cable. One is a Firewall
Router, connected to the DSL modem, the other is the new
Router with wireless access point. It replaces a Workgroup
switch and separate WiFi access point (which will be added
to the Firewall Router once the network is up and running).

Two WiFi access points will give better coverage than the
single one. Most likely any new connected appliances will be
wireless.

I can 'talk' to each router with a local PC, and both work
one at a time with the DSL modem. They don't work together.

I have changed the IP address of the new router from the
default (192.168.1.1) to 192.168.2.1, and can ping each
router one at a time.

Make sure that you have downloaded and installed the latest firmware update from Linksys. You may need to connect the router directly to a computer to update the firmware and for initial setup. Disconnect the other router temporarily.

Make sure the DHCP server is off on the wireless router. There can only be one DHCP server, two will fight each other and cause strange problems. On my Linksys, this setting is on the "Basic Setup" tab, about mid-way on the page.

Under "Advanced Routing", change the operating mode to "Router", instead of gateway.

Finally, on the wireless router, make sure that you're using the "uplink" port to connect the cable. On the other router, connect to a standard port. You won't be able to use port 1 on the wireless router, because of the uplink port.
EDIT: I forgot to add, make sure that you're using a standard cable, not a cross-over cable to connect the two routers!

With one router set up as 192.168.1.1 and the other as 192.168.2.1 you have configured two separate "networks" (address ranges) and making them talk to one another, while possible, will require some doing. I'd stick with just one if I had my druthers. In lay terms: keep everybody's IP address as 192.168.1.x, it's easier.

I have a setup similar to yours (or at least what I think you'd like yours to be). There are a number of ways it can work, here's the way mine does:

1) the (older, no wireless) router/firewall connected to my DSL modem has IP address 192.168.1.1 and is set up to be a DHCP server. Its uplink port is connected to my DSL modem. One of my servers is connected to one of its "internal" ports (it has 4) and takes advantage of the DMZ capabilities the router has. The router serves as my primary gateway/connection to the Internet and is configured with settings specific to my ISP (AT&T). Another one of its ports is connected to the router I describe in (2). I manage it by going to http://192.168.1.1 in a browser.

2) the wireless router/firewall is connected to the aforementioned router with a hard wire. The uplink port is not used. Port 1 of this router is connected to port 1 of the aforementioned router (though it doesn't matter which of the internal ports on either router one uses). DHCP is disabled in this router. Its local area network address is 192.168.1.15. The SSID and wireless security protocol (WPA) for my LAN are setup on this one. None of the Internet connection settings (ISP) are configured, leaving the uplink port unplugged renders them irrelevant anyway. To manage it, I point a browser at http://192.168.1.15.

So what's going on?

Well, the (wired) router described in (1) is my "firewall" in the traditional sense. It protects my home network from sinister elements of the Internet. Also, it manages assigning addresses to my PCs that are set up as DHCP clients and serves as the primary gateway for my LAN.

The wireless router, with its uplink port disconnected, is functioning as an "access point" (without paying an extra $50+ at the dealer for a box labeled as such). It is a repeater of sorts in the sense that any more cables plugged into its ports and any wireless connections made to it will act as though they are direct connections to the router described in (1). Since the firewall functions only act upon packets that pass between the 4 internal ports and the uplink port, this particular router does not perform any firewall tasks. However, it passes traffic sent to its ports and over wireless connections to the router described in (1) in a way similar to the way your trusty network switch passes traffic.

I have a WRT54GL set up as an access point at a client site and use one at home as well (with OpenWRT, a completely different animal that I use to manage my second (Comcast Workplace) connection). You paid extra for a router that one can do a lot with (see: Wifi Robot). They are considered to be one of the better consumer-grade devices.